Problem: Find the sum of the first $9$ terms in the following geometric series. Do not round your answer. $7+21+63+...$
Explanation: This formula gives the sum ${S_n}$ of the first $ n$ terms in the geometric series where the first term is $ a$ and the common ratio is $C r$ : ${S_n}=\dfrac{ a(1-C r^{ n})}{1-C r}$ We are given the value for $ n$. We are also given the first terms in the series, which tells us the values for, $ a$ and $C r$. After we find them, let's plug them in the formula. We are given that ${n=9}$. Since the series is $7+21+63+...$, we can tell that ${a=7}$, and $C{r=3}$. ${S_n}=\dfrac{ 7(1-C 3^{{9}})}{1-C 3}$ Evaluating the expression in the calculator, we get that $S_n=68{,}887$. In conclusion, the sum of the first $9$ terms in the series is $68{,}887$.